Orbital energies, hydrogen-like species

Besides providing information about the wavefunctions, solutions of the Schrodinger equation give orbital energies, E (energy levels), and equation 1.16 shows the dependence of E on the principal quantum number for hydrogen-like species. 

For each value of n there is only one energy solution and for hydrogen-like species, all atomic orbitals with the same principal quantum number (e.g. Zs, Zp and Zd) are degenerate. 

For a hydrogen-like species, confirm that the energy of the 3s orbital is -1.51 eV. 

Orbital energies in a hydrogen-like species Size of orbitals... 

The energy required to remove an electron from an orbital with principal quantum number n in a hydrogen-like species—the ionization energy—is the negative of that shown above, that is,... 

A helium +1 cation, like a hydrogen atom, has just one electron. Absorption and emission spectra show that He" has energy levels that depend on u, just like the hydrogen atom. Nevertheless, Figure 8 2 shows that the emission spectra of He and H differ, which means that these two species must have different energy levels. We conclude that something besides U influences orbital energy. 

In Chapter 2, the development of atomic orbitals for many-electron atoms was built upon an understanding of the orbitals obtained from the exact solution for the one-electron hydrogen atom. Similarly, it is useful to examine the electronic structure for the simplest molecular species—the hydrogen molecule ion (Hj)—to begin our discussion of molecular orbitals. Like the H atom, the ion contains only one electron. The difference is the presence of two nuclei instead of one. The potential energy of the system is a sum of the Coulomb attraction of the electron to each of the two nuclei and Coulomb repulsion between the positively charged nuclei. 

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